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 A323522 Number of ways to fill a square matrix with the parts of a strict integer partition of n. 4
 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25, 25, 49, 73, 121, 145, 217, 265, 361, 433, 553, 649, 817, 937, 1129, 1297, 1537, 1729, 2017, 2257, 2593, 2881, 3265, 3601, 4057, 4441, 4945, 5401, 5977, 6481, 7129, 7705, 8425, 9073, 9865, 373465, 374353, 738025, 1101865, 1828513 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,11 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..7000 FORMULA a(n) = Sum_{k >= 0} (k^2)! * Q(n, k^2) where Q = A008289. EXAMPLE The a(10) = 25 matrices:   [10] .   [4 3] [4 3] [4 2] [4 2] [4 1] [4 1] [3 4] [3 4]   [2 1] [1 2] [3 1] [1 3] [3 2] [2 3] [2 1] [1 2] .   [3 2] [3 2] [3 1] [3 1] [2 4] [2 4] [2 3] [2 3]   [4 1] [1 4] [4 2] [2 4] [3 1] [1 3] [4 1] [1 4] .   [2 1] [2 1] [1 4] [1 4] [1 3] [1 3] [1 2] [1 2]   [4 3] [3 4] [3 2] [2 3] [4 2] [2 4] [4 3] [3 4] MAPLE b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)       -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))     end: a:= n-> (l-> add(l[i^2+1]*(i^2)!, i=0..floor(sqrt(nops(l)-1))))(b(n\$2)): seq(a(n), n=0..50);  # Alois P. Heinz, Jan 17 2019 MATHEMATICA Table[Sum[(k^2)!*Length[Select[IntegerPartitions[n, {k^2}], UnsameQ@@#&]], {k, n}], {n, 20}] (* Second program: *) q[n_, k_] := q[n, k] = If[n < k || k < 1, 0,      If[n == 1, 1, q[n-k, k] + q[n-k, k-1]]]; a[n_] := If[n == 0, 1, Sum[(k^2)! q[n, k^2], {k, 0, n}]]; a /@ Range[0, 50] (* Jean-François Alcover, May 20 2021 *) CROSSREFS Cf. A000009, A089299, A103198 (non-strict case), A120732, A323431, A323434, A323519, A323523, A323525, A323529. Sequence in context: A283710 A247650 A090092 * A040601 A022359 A147491 Adjacent sequences:  A323519 A323520 A323521 * A323523 A323524 A323525 KEYWORD nonn AUTHOR Gus Wiseman, Jan 17 2019 STATUS approved

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Last modified September 20 05:37 EDT 2021. Contains 347577 sequences. (Running on oeis4.)